5 ms·

Inserting A=60 into the original line of reasoning (just to show that it doesn't immediately resolve the paradox): 1. Denote by A=60 the amount in the player's

by mrow84 4y ago

Inserting A=60 into the original line of reasoning (just to show that it doesn't immediately resolve the paradox): 1. Denote by A=60 the amount in the player's selected envelope. 2. The probability that A=60 is the smaller amount is 1/2, and that it is the larger amount is also 1/2. 3. The other envelope may contain either 2A=120 or A/2=30. 4. If A=60 is the smaller amount, then the other envelope contains 2A=120. 5. If A=60 is the larger amount, then the other envelope contains A/2=30. 6. Thus the other envelope contains 2A=120 with probability 1/2 and A/2=30 with probability 1/2. 7. So the expected value of the money in the other envelope is: 1/2(120) + 1/2(30) = 75 8. This is greater than A=60 so, on average, the person reasons that they stand to gain by swapping. 9. ... edit: If anything, to me this obscures things further, because it makes it non-obvious that we are actually talking about ratios (i.e. 2A, 1/2A) rather than absolute numbers, and so the expectation calculated in step 7 is inappropriate - a geometric mean recovers the correct answer.

JoshCole 4y ago
A can't collapse to sixty. It could still be 30, 60 or 120. The expected value calculation includes the value of A as defined in two different worlds: the world in which A/2 (30) and the world in which 2A (120). But these two worlds never exist together. They are different realities. We can't determine which of them we are in without more information. Trying to do so is an error, because we are in multiple subgames simultaneously. In imperfect information games you are already in more than one world. You are in the reality A and counterfactual 2A and you are in the reality 2A and counterfactual A. You can't tell which. You're in both. In perfect information, you're not in both, so you can do this case based reasoning and not run into as much trouble. But when you're already in both worlds? Well doing the case based reasoning implies that the counterfactual world = the case based world. Its overwriting that part of the equations. Which means you just accidentally declared that either A/2=A or that 2A=A. That 2=1. You do this because you're still in the counterfactual reality, but you're pretending you are not.
- mrow84 4y ago
  In the variant under discussion A is defined as 60: https://news.ycombinator.com/item?id=31574239 https://news.ycombinator.com/item?id=31574239
  - JoshCole 4y ago
    You don't know what A is even if he tells you the first envelope is 60. A and 2A exist together counterfactually, if you have A, you have 2A, if you have 2A, you have A. This is a fundamental property of imperfect information games. You don't have State=A, ever, and you are lying to yourself when you pretend that you do. It is /why/ the reasoning error happens. State based reasoning only works in perfect information games because in that framing you don't have counterfactual subgames forcing them to depend on each other - forcing you to play them both simultaneously, because you don't know which subgame you are in. A and 2A are together, so you can't propose A/2 without violating the A and 2A codependency imposed by not knowing which subgame you are in. There are multiple A when you get told 60. They are 30, 60, and 120. This leads to three potential solutions to the expected value: 3/2(30), 3/2(60), 3/2(120). You can't tell which of these solutions you are in, because you don't know. Regardless, since the terminating R(KEEP) condition is the only one that is defined P(SWITCH) = 1 still is either undetermined or 0 depending on how you solve the bellman equations. Your policy decides your expected value. So earlier I was a little loose when I claimed the 3/2A relationship. Honestly, the wikipedia article is really terrible. It demands you stick in its formalism and declares you a no true scotssman if you don't, buts its approach is fundamentally wrong. Throw off its chains and consider the framing where you remove the requirement of thinking about imperfect information. People are bad at it and if this is confusing you then it is /because/ you are struggling with the imperfect information. The way you convert between the two game types is this: Instead of states -> information sets. So you only have one move in this game. You always have the Null information set. You always have the same situation. So you're only allowed to have one choice and that is it every time forever. So you always have to make the same decision. Secondly instead of actions -> probability vectors over actions. To simplify and make it tractable assume [0.0, 0.1, 0.2, ... 1.0] so the action space is small. You are choosing 'an action' to take and its going to create multiple subgames. Focus on the recurrence relationship between those games. In particular look at the base case: R(KEEP) = {0.5: A, 0.5: 2A}. Everything heads toward that base case except one thing: P(SWITCH)=1. It is a markov process and R(SWITCH) 'drains' so as to be R(KEEP) because any probability in it inevitably becomes R(KEEP) after enough iterations. Notice that {0.5: A, 0.5: 2A} is always true! It is true for both you have A and you keep it and it is true for both you have 2A and you keep it, because notice - you never ever have A. You have the null information set. You can't ever see a difference between these two things. The moment you force in the idea that you can actually define A to be a particular thing, you smash all over that recurrence relationship. You destroy the relationship. You claim there is no relationship between the policy function and the expected value even though /there is/. There is so much wrong with replacing R(SWITCH) with a fake reality where you can you know you're actually in 1/2A when you can't. And that is what the equations are doing. They're saying you can replace the dependence on each other with a subgame that exists in a different reality. Lets say 120 was when you got 60 - there is no 30 in existence, but your equation calls to replace the recurrence relationship with the idea there is one. It doesn't make sense.
deleted 4y ago
[deleted]

by mrow84 4y ago

JoshCole 4y ago

A can't collapse to sixty. It could still be 30, 60 or 120. The expected value calculation includes the value of A as defined in two different worlds: the world in which A/2 (30) and the world in which 2A (120). But these two worlds never exist together. They are different realities. We can't determine which of them we are in without more information. Trying to do so is an error, because we are in multiple subgames simultaneously. In imperfect information games you are already in more than one world. You are in the reality A and counterfactual 2A and you are in the reality 2A and counterfactual A. You can't tell which. You're in both. In perfect information, you're not in both, so you can do this case based reasoning and not run into as much trouble. But when you're already in both worlds? Well doing the case based reasoning implies that the counterfactual world = the case based world. Its overwriting that part of the equations. Which means you just accidentally declared that either A/2=A or that 2A=A. That 2=1. You do this because you're still in the counterfactual reality, but you're pretending you are not.

mrow84 4y ago
In the variant under discussion A is defined as 60: https://news.ycombinator.com/item?id=31574239 https://news.ycombinator.com/item?id=31574239
- JoshCole 4y ago
  You don't know what A is even if he tells you the first envelope is 60. A and 2A exist together counterfactually, if you have A, you have 2A, if you have 2A, you have A. This is a fundamental property of imperfect information games. You don't have State=A, ever, and you are lying to yourself when you pretend that you do. It is /why/ the reasoning error happens. State based reasoning only works in perfect information games because in that framing you don't have counterfactual subgames forcing them to depend on each other - forcing you to play them both simultaneously, because you don't know which subgame you are in. A and 2A are together, so you can't propose A/2 without violating the A and 2A codependency imposed by not knowing which subgame you are in. There are multiple A when you get told 60. They are 30, 60, and 120. This leads to three potential solutions to the expected value: 3/2(30), 3/2(60), 3/2(120). You can't tell which of these solutions you are in, because you don't know. Regardless, since the terminating R(KEEP) condition is the only one that is defined P(SWITCH) = 1 still is either undetermined or 0 depending on how you solve the bellman equations. Your policy decides your expected value. So earlier I was a little loose when I claimed the 3/2A relationship. Honestly, the wikipedia article is really terrible. It demands you stick in its formalism and declares you a no true scotssman if you don't, buts its approach is fundamentally wrong. Throw off its chains and consider the framing where you remove the requirement of thinking about imperfect information. People are bad at it and if this is confusing you then it is /because/ you are struggling with the imperfect information. The way you convert between the two game types is this: Instead of states -> information sets. So you only have one move in this game. You always have the Null information set. You always have the same situation. So you're only allowed to have one choice and that is it every time forever. So you always have to make the same decision. Secondly instead of actions -> probability vectors over actions. To simplify and make it tractable assume [0.0, 0.1, 0.2, ... 1.0] so the action space is small. You are choosing 'an action' to take and its going to create multiple subgames. Focus on the recurrence relationship between those games. In particular look at the base case: R(KEEP) = {0.5: A, 0.5: 2A}. Everything heads toward that base case except one thing: P(SWITCH)=1. It is a markov process and R(SWITCH) 'drains' so as to be R(KEEP) because any probability in it inevitably becomes R(KEEP) after enough iterations. Notice that {0.5: A, 0.5: 2A} is always true! It is true for both you have A and you keep it and it is true for both you have 2A and you keep it, because notice - you never ever have A. You have the null information set. You can't ever see a difference between these two things. The moment you force in the idea that you can actually define A to be a particular thing, you smash all over that recurrence relationship. You destroy the relationship. You claim there is no relationship between the policy function and the expected value even though /there is/. There is so much wrong with replacing R(SWITCH) with a fake reality where you can you know you're actually in 1/2A when you can't. And that is what the equations are doing. They're saying you can replace the dependence on each other with a subgame that exists in a different reality. Lets say 120 was when you got 60 - there is no 30 in existence, but your equation calls to replace the recurrence relationship with the idea there is one. It doesn't make sense.

deleted 4y ago

[deleted]